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Calculus Archive: Questions from November 27, 2022

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  If \( f(x, y)=x \cos y+y e^{x} \), find \( \frac{\partial^{2} f}{\partial y \partial x} \) \[ \begin{array}{l} -\sin y+e^{x} \\ -\sin x+e^{y} \\ \sin y+e^{x} \\ -\cos x+e^{y} \end{array} \]
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• Which textbook/sources are these calculus questions from?
  In Problems 11-40 verify that the indicated function is a solution of the given differential equation. Where appropriate, \( c_{1} \) and \( c_{2} \) denote constants. 11. \( 2 y^{\prime}+y=0 ; \quad
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  1. Determine whether the following sequence converges or diverges. (a) \( \left\{\frac{3 n^{2}+10}{e^{n}}\right\} \) (b) \( \left\{\frac{1.3 .5 \ldots .(2 n-1)}{2.4 .6 \ldots .(2 n)}\right\} \) (c) \(
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  \( y=\frac{2+\sin x}{x+\cos x} \)
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• NEED HELP with the following problems (1-5) (Requires full procedure, not just selecting the right answer): 1. Need to find the convergence radius 2. Need to find the convergence inteval 3. Need to f
  1. El radio de convergencia de \( \sum_{n=1}^{\infty} \frac{x^{n}}{2^{n-1}} \) es: a. \( R=1 \) b. \( R=0 \) c. \( R=2 \) d. ninguna de las anteriores 2. El intervalo de convergencia de \( \sum_{n=1}^
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• 22. Choose the correct answer (show full procedure) Supose that an3^n converges Translation (see pictures for the actual excersise/problem since some symbols can't be typed and could create confusio
  22. Suponga que \( \sum_{n=1}^{\infty} a_{n} 3^{n} \) converge. a. (8\%) Bncuentre \( \lim _{n \rightarrow \infty}\left(a_{n} 3^{n}\right) \). Luego pruebe que \( \sqrt[2]{a_{n}} \leq \frac{1}{3} \) p
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• 𝑥=−(6𝑢+7𝑣), 𝑦=9𝑢+8𝑣 implies ∂(𝑢,𝑣)∂(𝑥,𝑦)=
  \( x=-(6 u+7 v), y=9 u+8 v \) implies \( \frac{\partial(u, v)}{\partial(x, y)}= \)
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  1. Find the solution for the following differential equations a. \( \quad\left(e^{y}-y e^{x}\right) d x+\left(x e^{y}-e^{x}\right) d y=0 \) b. \( \left(\sin y \cos y+x \cos ^{2} y\right) d x+x d y=0 \
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• Use the Winplot program and draw the graph f(x) 1/(x^2-2x-3) to tell whether the integral 0 to 2 f(x)dx is positive or negative. use the graph to give an estimate of the value of the integral, and th
  Resuelva: Use el programa Winplot y dibuje la gráfica de \( f(x)=\frac{1}{\left(x^{2}-2 x-3\right)} \bigcirc \) para decidir si \( \int_{0}^{2} f(x) d x \) es positiva o negativa. Use la gráfica par
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• Resolver la siguiente integral, utilizando la técnica más apropiada, integración por partes o tabulación.
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• please answer question 15
  9-20 Find the exact length of the curve. 9. \( y=1+6 x^{3 / 2}, \quad 0 \leqslant x \leqslant 1 \) 10. \( 36 y^{2}=\left(x^{2}-4\right)^{3}, \quad 2 \leqslant x \leqslant 3, \quad y \geqslant \) 11. \
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• NEED HELP with the following excersices (full procedure required):
  12. La superficie en el espacio dada por la ecuacion \( x^{2}+z^{2}=1 \) es: a. esfera unitaria b. cilindro c. elipse d. ninguna de las anteriores 13. Una representacion como serie de potencia de \( \
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  9. Find the asymptotes of the hyperbola \( 5 x^{2}-4 y^{2}=20 \) (a) \( y=\pm \frac{2}{\sqrt{5}} x \) (b) \( y=\pm \frac{5}{4} x \) (c) \( y=\pm \frac{\sqrt{5}}{2} x \) (d) \( y=\pm \frac{4}{5} x \)
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  (1 point) Calculate all four second-order partial derivatives of \( f(x, y)=\sin \left(\frac{4 x}{y}\right) \). \[ f_{x x}(x, y)= \] \[ f_{x y}(x, y)= \] \[ f_{y x}(x, y)= \] \[ f_{y y}(x, y) \]
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  Evaluate \( \iiint_{\mathcal{B}} f(x, y, z) d V \) for the specified function \( f \) and \( \mathcal{B} \) : \[ f(x, y, z)=\frac{z}{x} \quad 2 \leq x \leq 16,0 \leq y \leq 8,0 \leq z \leq 4 \] \[ \ii
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• Sketch the graph of f(x)=1/(x2-2x-3) to decide whether f(x)dx is positive or negative. Use the graph to give a rough estimate of the value of the integral, and then use partial fractions to find the e
  Use el programa Winplot y dibuje la gráfica de \( f(x)=\frac{1}{\left(x^{2}-2 x-z\right)} \bigcirc \) para decidir si \( \int_{0}^{2} f(x) d x \bigcirc \) es positiva o negativa. Use la gráica para
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  10. Given \( h(x)=f(g(x)) \) find \( h^{\prime}(2) \) if \( f(2)=3, \quad g(2)=4, \quad f^{\prime}(2)=1, \quad f^{\prime}(4)=5, \quad g^{\prime}(2)=6 \)
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  \( g(x) \) Invalid \( < \) msup \( > \) element \( =2 \) halle los intervalos donde la función es creciente o decreciente. a. La función es creciente en los intervalos \( (-\infty,-1) \) y \( (2, \i
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  Para la función \[ f(x)=x^{3}-6 x^{2}+9 x+1 \] en el intervalo cerrado \( [1,5] \) : I. Halle los números criticos de la función. II. Halle los valores máximos y mínimos (máximos y mínimos abso
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  For \( f(x, y, z)=3 x^{2} y-5 y^{2} z-4 x z^{2} \), find a. \( \lim _{h \rightarrow 0} \frac{f(x+h, y, z)-f(x, y, z)}{h} \) b. \( \lim _{h \rightarrow 0} \frac{f(x, y+h, z)-f(x, y, z)}{h} \) c. \( \li
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  2. El intervalo de convergencia de \( \sum_{n=1}^{\infty} \frac{x^{n}}{2^{n-1}} \) es: a. \( [-1 / 2,1 / 2] \) b. \( (-1 / 2,1 / 2) \) c. \( (-1 / 2,1 / 2] \) d. ninguna de las anteriores
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  4. La siguiente serie es absolutamente convergente: a. \( \sum_{n=1}^{\infty} \frac{(-1)^{n}}{\sqrt{n}} \) b. \( \sum_{n=1}^{\infty} \frac{(-1)^{n}}{n^{2}} \) c. \( \sum_{n=1}^{\infty} n \) d. ninguna
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  7. \( \sum_{n=1}^{\infty} \frac{1}{n^{p}} \) es divergente si a. \( p \leq 1 \) b. \( p>1 \) c. la a. y la b. son correctas d. ninguna de las anteriores
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  If \( \sum_{k=1}^{n} a_{k}=3 \) and \( \sum_{k=1}^{n} b_{k}=42 \), find the following values. \[ \sum_{k=1}^{n} 9 a_{k}, \quad \sum_{k=1}^{n} \frac{b_{k}}{42}, \sum_{k=1}^{n}\left(a_{k}+b_{k}\right),
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  7. Find \( y^{\prime \prime} \) if \( x^{2}+y^{2}=4 \). Simplify your answer. A. \( y^{\prime \prime}=\frac{2}{y^{2}} \) B. \( y^{\prime \prime}=\frac{2}{y^{3}} \) C. \( y^{\prime \prime}=-\frac{4}{y^
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• 3.22
  Evaluate \( \iiint_{E}(x+y-4 z) d V \) where \[ E=\left\{(x, y, z) \mid-4 \leq y \leq 0,0 \leq x \leq y, 0
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  \( f(x)=\sqrt{x} \cos x \), then \( f^{\prime}(x)= \) A) \( -\frac{\sin x}{2 \sqrt{x}} \) (B) \( \frac{1-2 \sqrt{x} \sin x}{2 \sqrt{x}} \) (C) \( \frac{\cos x-2 x \sin x}{2 \sqrt{x}} \) (D) \( \frac{\
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  1. En los siguientes ejercicios halle la longitud de arco en el intervalo dado: a. \( r(t)=t \boldsymbol{i}+3 t \boldsymbol{j}, \quad[0,4] \) b. \( r(t)=t^{3} \boldsymbol{i}+t^{2} \boldsymbol{j}, \qua
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  2. En los siguientes ejercicios hallar la curvatura \( K \) de cada una: a. \( r(t)=t \boldsymbol{i}+t^{2} \boldsymbol{j}+\frac{t^{2}}{2} \boldsymbol{k} \) b. \( r(t)=2 t^{2} \boldsymbol{i}+t \boldsym
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• Compute the gradient vector fields of the following functions:
  Section 13.1: Problem 1 (1 point) Compute the gradient vector fields of the following functions: A. \( f(x, y)=5 x^{2}+5 y^{2} \) \( \nabla f(x, y)=\mathbf{i}+\mathbf{j} \) B. \( f(x, y)=x^{5} y^{6} \
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  \( 1-10 \) * Evaluate the double integral. 7. \( \iint_{D} y^{2} d A, \quad D=\{(x, y) \mid-1 \leqslant y \leqslant 1,-y-2 \leqslant x \leqslant y\} \) 8. \( \iint_{D} \frac{y}{x^{5}+1} d A, \quad D=\
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  6. SOlve a) \( \int_{0}^{1} \int_{x}^{\sqrt{2-x^{2}}} 2 x+y d y d x \) b) \( \int_{0}^{2} \int_{0}^{\sqrt{4-y^{2}}} x d x d y \)
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  4. EVALUATE THE INTEGRALS. \[ \int_{0}^{\pi} \sin ^{4}(3 x) d x \]
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• please do 11,14,18
  In Problems 11 through 20, write the given system in the form \( \mathbf{x}^{\prime}=\mathbf{P}(t) \mathbf{x}+\mathbf{f}(t) \) 11. \( x^{\prime}=-3 y, y^{\prime}=3 x \) 12. \( x^{\prime}=3 x-2 y, y^{\
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• y = 2 cos(x), y = 2 − 2 cos(x), 0 ≤ x ≤ 𝜋
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  Calculate \( \iint_{\mathcal{S}} f(x, y, z) d S \) For Part of the surface \( x=z^{3} \), where \( 0 \leq x, y \leq 4^{-\frac{3}{2}} ; \quad f(x, y, z)=x \) \[ \iint_{\mathcal{S}} f(x, y, z) d S= \]
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  \( y=\ln \left(7 x^{5}+x^{3}-5 x\right) \) \( y=\left(2 x^{3}-5 x\right) \) \( y^{\prime} \) if \( y=\left(x^{3}+2 x\right) e^{-5 x} \)
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  \( y^{\prime} \) if \( y=\frac{x^{3}}{e^{2 x}} \) \( y^{\prime} \) if \( y=\ln e^{\left(x^{2}+3 x\right)} \) \( y^{\prime} \) if \( y=e^{-4 x} \ln 5 x \) \( y^{\prime} \) if \( y=x^{\left(x^{2}+3 x\ri
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• Find the exact length of the curve. x = 1 3 y (y − 3), 9 ≤ y ≤ 25
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